Periodic Gamma Autoregressive Processes for Operational Hydrology

Abstract

A number of models have been suggested •or hydrologic time series in general and streamflow series in particular. Most of them are normal autoregressive (AR) of order 1 with either constant or periodic parameters. Since generally hydrologic time series are nonnormal (skewed), transformations have been suggested to make the series approximately normal. A new class of univariate models is proposed herein which incorporates skewed and correlation properties within the model structure without the necessity of transformations. Such models assume a gamma marginal distribution and a constant or periodic aut0regressive structure. The models may be additive gamma, multiplicative. gamma, or a mixed model which incorporates properties of both additive and multiplicative models. The gamma models were tested and compared in relation to (transformed) normal AR models by computer simulation studies based on five weekly streamflow series with samples varying from 35 to 40 years of record. The results show that the new class of gamma models compares favorably with respect to the normal models in reproducing the basic statistics usually analyzed for streambow simulation. It is expected that the proposed gamma models will be of interest to other researchers for further developments and applications to hydrologic and geophysical time series. INTRODUCTION Synthetic generation of streamflow sequence_s is commonly utilized in planning and operational studieg of water resources systems. Such synthetic sequences are generated so as to rep•roduce a set of statistical properties which are found in the historical streamflow data. For such purpose, a number of time series models have been suggested since Hennan [1955] and Thomas and Fiering [1962] applied G•tussian auto•egressire (AR) models to seasonal hydrologic series. In modeling these series it is usually assumed that they are seasonally stationary and that seasonality is reflected in the mean, variance, covariance, and the skewness. However, in general, one may assume seasonality in the marginal distribution. Annual streamflow series may be considered approximately normal' however, for shorter time interval series such as monthly, weekly, and daily series, departures from normality become important. The problem of skewed streamflows has been handled by a number of methods. A Widely used technique is to use transformations to render a series close to normal [Box and Cox, 1965]. Examples of this approach are given by Matalas [1967], Mejia and Rodri•iuez-lturbe [1974], Lettenrnaier and Burqes [1977], Hirsch [1979], and Stedinqer [1981]. Another approach is to find the statistical properties of the noise in such a way as to reproduce the skew of the series. See, for instance, the Wilson-Hilferty (W-H) transformations used by Thomas and Fierinq [1962], Beard [1965], Payne et al. [1969], McGinnis and Sammens [1970], McMahon and Miller [1971], and O'Connell [1974]. Although the coefficient of skewness can be reproduced by this approach, the underlying variable is not gamma. A related problem ...

A Cautionary Note on the Reproduction of Dependencies through Linear Stochastic Models with Non-Gaussian White Noise

“…Nevertheless, it is noteworthy that Song et al [45] and Jeong and Lee [46] also observed this issue independently while studying AR(1) with exponential white noise [47][48][49] and periodic Gamma autoregressive (PGAR) processes [50], respectively. However, to the best of our knowledge, these works, or any other, have not revealed the envelope limitation, neither provided a theoretical proof and a justification for this behavior, which probably arises from the lack of explicit assumption regarding the joint dependence structure of the process.…”

Multivariate synthetic streamflow generation using a hybrid model based on artificial neural networks

“…Classical nonlinear approaches typically require large amounts of exogenous information, which is not always available (Deo and Thirumalaiah, 2000). Some nonlinear and non-Gaussian techniques do not need exogenous information and behave better than linear models, as in the case of periodic gamma autoregressive processes PGAR (Fernandez and Salas, 1986), but they are univariate models. Different authors (Lapedes and Farber, 1988;Tang et al, 1991;Zealand et al, 1999;Imrie et al, 2000;and Salas et al, 2000) have tested the capability of certain NN topologies to incorporate complex and non-linear hydrological relationships; they remark on their potentials and abilities as tools for hydrological forecasting.…”

“…This comparison is the right way to validate a model if it is intended for water resources systems planning and management management and planning (Jackson, 1975;Salas et al, 1980;Stedinger and Taylor, 1982;Fernandez and Salas, 1986;Kendal and Dracup, 1991;Basson and van Rooyen, 2001). Therefore, considering that the proposed model is not a forecasting approach, the performance of the NN-based model must not be evaluated by using the classical procedure of splitting all available data into training and validation (and/or) test sets.…”

SOME SIMPLE MODELS FOR CONTINUOUS VARIATE TIME SERIES1

“…However, using the mixture scheme given above, this four-parameter second-order process can be extended to higherorders quite simply (Lawrance and Lewis, 1985a Fernandez and Salas (1985) have extended the autoregressive Gamma process to the case of periodic parameters; this is an extension to the nonGaussian case of the so-called "Thomas-Fiering model", although the model is actually due to Hannan (1955).…”